x / (x^2 + 1)

Percentage Accurate: 76.9% → 100.0%

Time: 2.2s

Alternatives: 5

Speedup: 0.9×

Specification

Math

\[\begin{array}{l} \\ \frac{x}{x \cdot x + 1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))

C

double code(double x) {
	return x / ((x * x) + 1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x / ((x * x) + 1.0d0)
end function

Java

public static double code(double x) {
	return x / ((x * x) + 1.0);
}

Python

def code(x):
	return x / ((x * x) + 1.0)

Julia

function code(x)
	return Float64(x / Float64(Float64(x * x) + 1.0))
end

MATLAB

function tmp = code(x)
	tmp = x / ((x * x) + 1.0);
end

Wolfram

code[x_] := N[(x / N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{x \cdot x + 1} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ x (+ (* x x) 1.0)))

C

double code(double x) {
	return x / ((x * x) + 1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x / ((x * x) + 1.0d0)
end function

Java

public static double code(double x) {
	return x / ((x * x) + 1.0);
}

Python

def code(x):
	return x / ((x * x) + 1.0)

Julia

function code(x)
	return Float64(x / Float64(Float64(x * x) + 1.0))
end

MATLAB

function tmp = code(x)
	tmp = x / ((x * x) + 1.0);
end

Wolfram

code[x_] := N[(x / N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 50000:\\ \;\;\;\;x\_m \cdot {\left(\mathsf{hypot}\left(1, x\_m\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m} - {x\_m}^{-3}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 50000.0)
    (* x_m (pow (hypot 1.0 x_m) -2.0))
    (- (/ 1.0 x_m) (pow x_m -3.0)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 50000.0) {
		tmp = x_m * pow(hypot(1.0, x_m), -2.0);
	} else {
		tmp = (1.0 / x_m) - pow(x_m, -3.0);
	}
	return x_s * tmp;
}

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 50000.0) {
		tmp = x_m * Math.pow(Math.hypot(1.0, x_m), -2.0);
	} else {
		tmp = (1.0 / x_m) - Math.pow(x_m, -3.0);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 50000.0:
		tmp = x_m * math.pow(math.hypot(1.0, x_m), -2.0)
	else:
		tmp = (1.0 / x_m) - math.pow(x_m, -3.0)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 50000.0)
		tmp = Float64(x_m * (hypot(1.0, x_m) ^ -2.0));
	else
		tmp = Float64(Float64(1.0 / x_m) - (x_m ^ -3.0));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 50000.0)
		tmp = x_m * (hypot(1.0, x_m) ^ -2.0);
	else
		tmp = (1.0 / x_m) - (x_m ^ -3.0);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 50000.0], N[(x$95$m * N[Power[N[Sqrt[1.0 ^ 2 + x$95$m ^ 2], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x$95$m), $MachinePrecision] - N[Power[x$95$m, -3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5e4

   1. Initial program 82.0%

      \[\frac{x}{x \cdot x + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 81.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + 1}{x}}} \]

      2. associate-/r/ 82.0%

         \[\leadsto \color{blue}{\frac{1}{x \cdot x + 1} \cdot x} \]

      3. add-sqr-sqrt 81.9%

         \[\leadsto \frac{1}{\color{blue}{\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}} \cdot x \]

      4. pow2 81.9%

         \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt{x \cdot x + 1}\right)}^{2}}} \cdot x \]

      5. pow-flip 82.0%

         \[\leadsto \color{blue}{{\left(\sqrt{x \cdot x + 1}\right)}^{\left(-2\right)}} \cdot x \]

      6. +-commutative 82.0%

         \[\leadsto {\left(\sqrt{\color{blue}{1 + x \cdot x}}\right)}^{\left(-2\right)} \cdot x \]

      7. hypot-1-def 83.1%

         \[\leadsto {\color{blue}{\left(\mathsf{hypot}\left(1, x\right)\right)}}^{\left(-2\right)} \cdot x \]

      8. metadata-eval 83.1%

         \[\leadsto {\left(\mathsf{hypot}\left(1, x\right)\right)}^{\color{blue}{-2}} \cdot x \]

   4. Applied egg-rr 83.1%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{-2} \cdot x} \]

   if 5e4 < x

   1. Initial program 62.9%

      \[\frac{x}{x \cdot x + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 63.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + 1}{x}}} \]

      2. associate-/r/ 62.7%

         \[\leadsto \color{blue}{\frac{1}{x \cdot x + 1} \cdot x} \]

      3. add-sqr-sqrt 62.7%

         \[\leadsto \frac{1}{\color{blue}{\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}} \cdot x \]

      4. pow2 62.7%

         \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt{x \cdot x + 1}\right)}^{2}}} \cdot x \]

      5. pow-flip 62.8%

         \[\leadsto \color{blue}{{\left(\sqrt{x \cdot x + 1}\right)}^{\left(-2\right)}} \cdot x \]

      6. +-commutative 62.8%

         \[\leadsto {\left(\sqrt{\color{blue}{1 + x \cdot x}}\right)}^{\left(-2\right)} \cdot x \]

      7. hypot-1-def 65.4%

         \[\leadsto {\color{blue}{\left(\mathsf{hypot}\left(1, x\right)\right)}}^{\left(-2\right)} \cdot x \]

      8. metadata-eval 65.4%

         \[\leadsto {\left(\mathsf{hypot}\left(1, x\right)\right)}^{\color{blue}{-2}} \cdot x \]

   4. Applied egg-rr 65.4%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{-2} \cdot x} \]

   5. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\frac{1}{x} - \frac{1}{{x}^{3}}} \]
   6. Step-by-step derivation

      1. exp-to-pow 100.0%

         \[\leadsto \frac{1}{x} - \frac{1}{\color{blue}{e^{\log x \cdot 3}}} \]

      2. *-commutative 100.0%

         \[\leadsto \frac{1}{x} - \frac{1}{e^{\color{blue}{3 \cdot \log x}}} \]

      3. exp-neg 100.0%

         \[\leadsto \frac{1}{x} - \color{blue}{e^{-3 \cdot \log x}} \]

      4. distribute-lft-neg-in 100.0%

         \[\leadsto \frac{1}{x} - e^{\color{blue}{\left(-3\right) \cdot \log x}} \]

      5. metadata-eval 100.0%

         \[\leadsto \frac{1}{x} - e^{\color{blue}{-3} \cdot \log x} \]

      6. log-pow 100.0%

         \[\leadsto \frac{1}{x} - e^{\color{blue}{\log \left({x}^{-3}\right)}} \]

      7. rem-exp-log 100.0%

         \[\leadsto \frac{1}{x} - \color{blue}{{x}^{-3}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{x} - {x}^{-3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 50000:\\ \;\;\;\;x \cdot {\left(\mathsf{hypot}\left(1, x\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 100000:\\ \;\;\;\;\frac{x\_m}{1 + x\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m} - {x\_m}^{-3}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 100000.0)
    (/ x_m (+ 1.0 (* x_m x_m)))
    (- (/ 1.0 x_m) (pow x_m -3.0)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 100000.0) {
		tmp = x_m / (1.0 + (x_m * x_m));
	} else {
		tmp = (1.0 / x_m) - pow(x_m, -3.0);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 100000.0d0) then
        tmp = x_m / (1.0d0 + (x_m * x_m))
    else
        tmp = (1.0d0 / x_m) - (x_m ** (-3.0d0))
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 100000.0) {
		tmp = x_m / (1.0 + (x_m * x_m));
	} else {
		tmp = (1.0 / x_m) - Math.pow(x_m, -3.0);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 100000.0:
		tmp = x_m / (1.0 + (x_m * x_m))
	else:
		tmp = (1.0 / x_m) - math.pow(x_m, -3.0)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 100000.0)
		tmp = Float64(x_m / Float64(1.0 + Float64(x_m * x_m)));
	else
		tmp = Float64(Float64(1.0 / x_m) - (x_m ^ -3.0));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 100000.0)
		tmp = x_m / (1.0 + (x_m * x_m));
	else
		tmp = (1.0 / x_m) - (x_m ^ -3.0);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 100000.0], N[(x$95$m / N[(1.0 + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x$95$m), $MachinePrecision] - N[Power[x$95$m, -3.0], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1e5

   1. Initial program 82.0%

      \[\frac{x}{x \cdot x + 1} \]
   2. Add Preprocessing

   if 1e5 < x

   1. Initial program 62.9%

      \[\frac{x}{x \cdot x + 1} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 63.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + 1}{x}}} \]

      2. associate-/r/ 62.7%

         \[\leadsto \color{blue}{\frac{1}{x \cdot x + 1} \cdot x} \]

      3. add-sqr-sqrt 62.7%

         \[\leadsto \frac{1}{\color{blue}{\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}} \cdot x \]

      4. pow2 62.7%

         \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt{x \cdot x + 1}\right)}^{2}}} \cdot x \]

      5. pow-flip 62.8%

         \[\leadsto \color{blue}{{\left(\sqrt{x \cdot x + 1}\right)}^{\left(-2\right)}} \cdot x \]

      6. +-commutative 62.8%

         \[\leadsto {\left(\sqrt{\color{blue}{1 + x \cdot x}}\right)}^{\left(-2\right)} \cdot x \]

      7. hypot-1-def 65.4%

         \[\leadsto {\color{blue}{\left(\mathsf{hypot}\left(1, x\right)\right)}}^{\left(-2\right)} \cdot x \]

      8. metadata-eval 65.4%

         \[\leadsto {\left(\mathsf{hypot}\left(1, x\right)\right)}^{\color{blue}{-2}} \cdot x \]

   4. Applied egg-rr 65.4%

      \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(1, x\right)\right)}^{-2} \cdot x} \]

   5. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\frac{1}{x} - \frac{1}{{x}^{3}}} \]
   6. Step-by-step derivation

      1. exp-to-pow 100.0%

         \[\leadsto \frac{1}{x} - \frac{1}{\color{blue}{e^{\log x \cdot 3}}} \]

      2. *-commutative 100.0%

         \[\leadsto \frac{1}{x} - \frac{1}{e^{\color{blue}{3 \cdot \log x}}} \]

      3. exp-neg 100.0%

         \[\leadsto \frac{1}{x} - \color{blue}{e^{-3 \cdot \log x}} \]

      4. distribute-lft-neg-in 100.0%

         \[\leadsto \frac{1}{x} - e^{\color{blue}{\left(-3\right) \cdot \log x}} \]

      5. metadata-eval 100.0%

         \[\leadsto \frac{1}{x} - e^{\color{blue}{-3} \cdot \log x} \]

      6. log-pow 100.0%

         \[\leadsto \frac{1}{x} - e^{\color{blue}{\log \left({x}^{-3}\right)}} \]

      7. rem-exp-log 100.0%

         \[\leadsto \frac{1}{x} - \color{blue}{{x}^{-3}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{x} - {x}^{-3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 100000:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - {x}^{-3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 50000000:\\ \;\;\;\;\frac{x\_m}{1 + x\_m \cdot x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (if (<= x_m 50000000.0) (/ x_m (+ 1.0 (* x_m x_m))) (/ 1.0 x_m))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 50000000.0) {
		tmp = x_m / (1.0 + (x_m * x_m));
	} else {
		tmp = 1.0 / x_m;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 50000000.0d0) then
        tmp = x_m / (1.0d0 + (x_m * x_m))
    else
        tmp = 1.0d0 / x_m
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 50000000.0) {
		tmp = x_m / (1.0 + (x_m * x_m));
	} else {
		tmp = 1.0 / x_m;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 50000000.0:
		tmp = x_m / (1.0 + (x_m * x_m))
	else:
		tmp = 1.0 / x_m
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 50000000.0)
		tmp = Float64(x_m / Float64(1.0 + Float64(x_m * x_m)));
	else
		tmp = Float64(1.0 / x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 50000000.0)
		tmp = x_m / (1.0 + (x_m * x_m));
	else
		tmp = 1.0 / x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 50000000.0], N[(x$95$m / N[(1.0 + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5e7

   1. Initial program 82.3%

      \[\frac{x}{x \cdot x + 1} \]
   2. Add Preprocessing

   if 5e7 < x

   1. Initial program 60.6%

      \[\frac{x}{x \cdot x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\frac{1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 50000000:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (if (<= x_m 1.0) x_m (/ 1.0 x_m))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = x_m;
	} else {
		tmp = 1.0 / x_m;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.0d0) then
        tmp = x_m
    else
        tmp = 1.0d0 / x_m
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 1.0) {
		tmp = x_m;
	} else {
		tmp = 1.0 / x_m;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 1.0:
		tmp = x_m
	else:
		tmp = 1.0 / x_m
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 1.0)
		tmp = x_m;
	else
		tmp = Float64(1.0 / x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 1.0)
		tmp = x_m;
	else
		tmp = 1.0 / x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 1.0], x$95$m, N[(1.0 / x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 81.9%

      \[\frac{x}{x \cdot x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 60.6%

      \[\leadsto \color{blue}{x} \]

   if 1 < x

   1. Initial program 63.4%

      \[\frac{x}{x \cdot x + 1} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 98.3%

      \[\leadsto \color{blue}{\frac{1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.5% accurate, 7.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot x\_m \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m) :precision binary64 (* x_s x_m))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * x_m;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * x_m
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * x_m;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * x_m

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * x_m)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * x_m;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * x$95$m), $MachinePrecision]

Derivation

1. Initial program 76.9%

   \[\frac{x}{x \cdot x + 1} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 45.5%

   \[\leadsto \color{blue}{x} \]

4. Final simplification 45.5%

   \[\leadsto x \]
5. Add Preprocessing

Developer target: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x + \frac{1}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (+ x (/ 1.0 x))))

C

double code(double x) {
	return 1.0 / (x + (1.0 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (x + (1.0d0 / x))
end function

Java

public static double code(double x) {
	return 1.0 / (x + (1.0 / x));
}

Python

def code(x):
	return 1.0 / (x + (1.0 / x))

Julia

function code(x)
	return Float64(1.0 / Float64(x + Float64(1.0 / x)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (x + (1.0 / x));
end

Wolfram

code[x_] := N[(1.0 / N[(x + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024036 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

  :herbie-target
  (/ 1.0 (+ x (/ 1.0 x)))

  (/ x (+ (* x x) 1.0)))